Several remarks on norm attaining in tensor product spaces

TL;DR

This paper investigates the conditions under which the norm in tensor product spaces is strongly subdifferentiable, revealing new properties of operator compactness and norm attainment in these spaces.

Contribution

It establishes new results on strong subdifferentiability of tensor product norms and the $w^*$-Kadec-Klee property, extending previous work and providing examples of dense norm-attaining tensors.

Findings

Strong subdifferentiability of tensor product norms under certain conditions

The $w^*$-Kadec-Klee property holds for duals of specific tensor products

Existence of spaces with dense but not unique norm-attaining tensors

Abstract

The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if $X\hat{\otimes}_\pi Y$ is strongly subdifferentiable and either $X$ or $Y$ has the metric approximation property then every bounded operator from $X$ to $Y^*$ is compact. We also prove that $(\ell_p(I)\hat{\otimes}_\pi \ell_q(J))^*$ has the $w^*$-Kadec-Klee property for every non-empty sets $I,J$ and every $2<p,q<\infty$, obtaining in particular that the norm of the space $\ell_p(I)\hat{\otimes}_\pi \ell_q(J)$ is strongly subdifferentiable. This extends several results of Dantas, Kim, Lee and Mazzitelli. We also find examples of spaces $X$ and $Y$ for which the set of norm-attaining tensors in $X\pten Y$ is dense but whose complement is dense too.